06_PlateTheory_08_Vibrations

06_PlateTheory_08_Vibrations - Section 6.8 Solid Mechanics...

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Unformatted text preview: Section 6.8 Solid Mechanics Part II Kelly 173 6.8 Plate Vibrations In this section, the problem of a vibrating circular plate will be considered. Vibrating plates will be re-examined again in the next section, using a strain energy formulation. 6.8.1 Vibrations of a Clamped Circular Plate When a plate vibrates with velocity t / , the third equation of equilibrium, Eqn. 6.6.2c becomes the equation of motion 2 2 t w z y x zz yz xz = + + (6.8.1) With this adjustment, the term q is replaced with 2 2 / t w h q + in the relevant equations; the acceleration term is treated as a transverse load of intensity 2 2 / t w h . Regarding the circular plate, one has from the axisymmetric governing equation 6.6.10 (with = q ), 2 2 2 2 2 1 t w D h w dr d r dr d = + (6.8.2) Assume a solution of the form ( ) ( ) + = t r W t r w cos ) , ( (6.8.3) Substituting into 6.8.2 gives 1 4 2 2 2 = + W k dr d r dr d (6.8.4) where D h k = 2 (6.8.5) Eqn. 6.8.4 gives the two differential equations 1 , 1 2 2 2 2 2 2 = + = + + W k dr d r dr d W k dr d r dr d (6.8.6) The solution to these equations are Section 6.8 Solid Mechanics Part II Kelly 174 ( ) ( ) ( ) ( ) kr K C kr I C W kr Y C kr J C W 4 3 2 1 , + = + = (6.8.7) where J and Y are, respectively, the Bessel functions of order zero of the first kind and...
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06_PlateTheory_08_Vibrations - Section 6.8 Solid Mechanics...

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